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Find the least square number which is exactly divisible by each of the numbers : 88, 1212, 1515,and 2020.

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Full solution

Q. Find the least square number which is exactly divisible by each of the numbers : 88, 1212, 1515,and 2020.
  1. Find LCM: To find the least square number that is exactly divisible by 88, 1212, 1515, and 2020, we first need to find the least common multiple (LCM) of these numbers. The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set.
  2. Prime Factorization: We start by finding the prime factorization of each number.\newline8=238 = 2^3\newline12=22×312 = 2^2 \times 3\newline15=3×515 = 3 \times 5\newline20=22×520 = 2^2 \times 5
  3. Calculate LCM: Next, we take the highest powers of all prime factors found in the factorization of each number to find the LCM. \newlineLCM = 23×3×52^3 \times 3 \times 5 (since 232^3 is the highest power of 22, 33 is the highest power of 33, and 55 is the highest power of 55)
  4. Ensure Square Number: Calculating the LCM gives us:\newlineLCM = 8×3×5=24×5=1208 \times 3 \times 5 = 24 \times 5 = 120
  5. Make LCM Square: Now, we need to ensure that the LCM is a square number. A square number is an integer that is the square of another integer. If the LCM is not a square number, we need to multiply it by the smallest number necessary to make it a square.
  6. Multiply by Factors: The prime factorization of the LCM 120120 is 23×3×52^3 \times 3 \times 5. To make it a square, we need to have even powers of all prime factors. Currently, the powers of 22, 33, and 55 are not even.
  7. Final Result: To make the LCM a square, we need to multiply it by another 22 (to make the power of 22 even), by 33 (to make the power of 33 even), and by 55 (to make the power of 55 even). This will give us the least square number.
  8. Final Result: To make the LCM a square, we need to multiply it by another 22 (to make the power of 22 even), by 33 (to make the power of 33 even), and by 55 (to make the power of 55 even). This will give us the least square number.Multiplying the LCM by 22, 33, and 55 gives us:\newlineLeast square number = LCM ×2×3×5=120×2×3×5=120×30=3600\times 2 \times 3 \times 5 = 120 \times 2 \times 3 \times 5 = 120 \times 30 = 3600
  9. Final Result: To make the LCM a square, we need to multiply it by another 22 (to make the power of 22 even), by 33 (to make the power of 33 even), and by 55 (to make the power of 55 even). This will give us the least square number.Multiplying the LCM by 22, 33, and 55 gives us:\newlineLeast square number = LCM ×2×3×5=120×2×3×5=120×30=3600\times 2 \times 3 \times 5 = 120 \times 2 \times 3 \times 5 = 120 \times 30 = 3600The prime factorization of 2200 is 2211, which is indeed a square number (2222). Therefore, 2200 is the least square number that is exactly divisible by 2244, 2255, 2266, and 2277.

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